In this work, we introduce non-Gaussian generalizations of the EnKF by
considering nonlinear transformations derived from couplings.
Couplings provide a link between distributions. That is, for a pair
of distributions \((\pi _1,\pi _2)\) , a coupling is defined by a pair of
random variables
\((X_1, X_2)\) , which admit \(\pi _1\)  and \(\pi _2\)  as
marginal distributions [1]}.
We consider couplings that are induced by a continuous transformation
\(T\)  such that \(X_2 = T(X_1)\) . The transformation can be either
deterministic or stochastic. When \(T\)  is deterministic, we call it a
transport map. The transformation allows one to sample \(\pi _2\) 
by evaluating \(T\)  at samples from \(\pi _1\) .
